Did you first erase the whole physical SSD as GUID partition and APFS (top option)? Within Disk Utility you need to click "View" and select "Show All Devices" so the physical SSD appears on the left pane of Disk Utility. The physical SSD will be identified something like "Apple SSD .....".
How are you booting the macOS installer? Are you booting into Recovery Mode (Command + R) or from a bootable macOS 10.15 USB installer?
Does your old SSD still work & boot? If so, or if you have access to another Mac from 2012 to 2019/2020 then I recommend creating and using a bootable macOS USB installer:
How to create a bootable installer for macOS - Apple Support
If you have problems with the USB installer, then try using another USB stick (especially a different brand) since the quality of USB sticks is extremely poor and Macs can be very picky about the drives used for booting.
As for your TM backup restore issue, that seems self explanatory. Your laptop is running an older OS than what is included in the backup. You can only restore a TM backup to the same version of macOS or later.
Try running the Apple Diagnostics to see if any hardware issues are detected. Just because the SSD fits into the SSD slot on the Logic Board does not necessarily mean the SSD will work with macOS on that particular Mac. If the Apple SSD you purchased was not designed to work with this 2015 Mac, then the diagnostics may report an issue and macOS may not work properly. In fact it is possible the SSD may only work in the 15" model and not the 13" model (or vice versa). I'm not saying for sure this is the case at all, but I do know that the Apple hardware, macOS, and Apple Diagnostics do care about these things sometimes even though a non-Apple OS like Linux may work perfectly well with the SSD in this laptop. This is from first hand personal experience.
Try an SMC Reset and PRAM Reset (hold the PRAM Reset for at least three chimes). In fact I would even recommend performing the SMC Reset first, then as soon as the laptop powers back on....immediately initiate the PRAM Reset. I once had to do this with back to back SMC & PRAM resets to get a third party SSD to work on an early MBAir. Individual resets did not allow the third party SSD to work. Yes, Macs can be quite cantankerous.
